magnetotelluric tensor decomposition: insights from linear...

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Magnetotelluric Tensor Decomposition: Insightsfrom Linear Algebra and Mohr Diagrams

F.E.M.(Ted) LilleyResearch School of Earth Sciences, Australian National University, Canberra

Australia

1. Introduction

The magnetotelluric (MT) method of geophysics exploits the phenomenon of naturalelectromagnetic induction which takes place at and near the surface of Earth. The purposeis to determine information about the electrical conductivity structure of Earth, upon whichthe process of electromagnetic induction depends. The MT method has been well describedrecently in books such as Simpson & Bahr (2005), Gubbins & Herrero-Bervera (2007), andBerdichevsky & Dmitriev (2008). The reader is referred to these for general information aboutthe method and its results. Notable modern extensions of the method are observation at anarray of sites simultaneously, and observation on the seafloor (both shallow and deep oceans).

In the most simple form of the method, data are observed at a single field site. Typically,three components (north, east and vertically downwards) of the fluctuating magnetic fieldare observed, and two components (north and east) of the fluctuating electric field. Themagnetic field is measured using a variety of instruments such as fluxgates and inductioncoils. The electric field is measured more simply, between grounded electrodes typicallyseveral hundred metres apart.

The natural signals observed cover a frequency band from 0.001 to 1000 Hz. They have avariety of causes, the relative importance of which varies with position on the Earth, especiallylatitude. Recorded data are transformed to the frequency domain, and interpretation proceedsbased on frequency-dependence.

The reduction of observed time-series to the frequency domain is thus fundamental to the MTmethod. In the frequency domain various transfer functions are determined, encapsulatingthe response of the observing site to the source fields causing the induction.

Another fundamental part of the data-reduction process, the focus of the present chapter, is“rotation” of observed data. By rotation is meant an examination of the observed transferfunctions to see how they would vary were the observing axes rotated at the observing site.Rotation may reveal geologic dimensionality, and strike direction.

The MT tensor has a 2 x 2 form and is well-suited to analysis by the methods of linear algebra.This suitability was evident early in the development of MT, and it is common for a paper onMT to start with such an analysis. The papers of Eggers (1982), LaTorraca et al. (1986), andYee & Paulson (1987), for example, specifically proceed with eigenvalue analysis and singularvalue decomposition (SVD), and see also Weaver (1994).

4

2 Will-be-set-by-IN-TECH

There appear however to be a number of aspects of this approach as yet unexplored, especiallywhen the in-phase and quadrature parts of the tensor are analysed separately. This chapternow follows such a line of enquiry, investigating the significance of symmetric, antisymmetricand non-symmetric parts, eigenvalue analysis and SVD.

The Mohr diagram representation is found to be a useful way to display some of the results.Cases of 1D, 2D and 3D electrical conductivity structure are examined, including the 3D casewhere the MT phase is greater than 90◦ or “out of quadrant”.

Taking the Smith (1995) treatment of telluric distortion, a particular case examined is thatwhere SVD gives an angle which is frequency independent, implying that it containsgeological information. Such angles arise when MT tensor data are nearly singular.

The analysis of in-phase and quadrature parts separately is seen to give further insight intothe question of whether, as suspected by Lilley (1998), the determinants of the parts takenseparately should always be positive.

2. Notation

The common representation of a magnetotelluric tensor Z is taken,

E = ZH (1)

of components [ExEy

]=

[Zxx ZxyZyx Zyy

] [HxHy

](2)

linking observed electric E and magnetic H fluctuations at an observing site on the surface ofEarth.

All quantities are complex functions of frequency ω, and in Equations 1 and 2 a timedependence of exp(iωt) is understood.

In this chapter the subscripts p, q will be used to denote in-phase and quadrature parts. Forexample the complex quantity Zσ is expressed

Zσ = Zσp + iZσ

q (3)

Note that adopting a time-dependence of exp(−iωt) as recommended by authors such asStratton (1941) and Hobbs (1992) would change the sign of Zσ

q . Such a change may bemisinterpreted, especially when distortion has caused the phase to be out of its expected firstquadrant.

The subscripts p, q will also be used to denote quantities which are derived from the in-phaseand quadrature parts, respectively, of a complex quantity, but which are themselves notrecombined to give a further complex quantity. In Sections 2.1, 3, 5 and 11 below, wherederivations apply equally to in-phase and quadrature cases, subscripts p, q are omitted forsimplicity. Such derivations can be taken as applying to in-phase components, with similarderivations possible for quadrature components.

Also in this chapter, for compactness of text, a 2 x 2 matrix such as that for Z in Equation 2will in places be written [Zxx, Zxy; Zyx, Zyy]. A rotation matrix R(θ) will be introduced

R(θ) = [cos θ, sin θ;− sin θ, cos θ] (4)

82 New Achievements in Geoscience

Magnetotelluric Tensor Decomposition: Insights from Linear Algebra and Mohr Diagrams 3

O

XX’

Y

Y’

θ’

Fig. 1. The rotation of MT observing axes clockwise by angle θ′, from OX and OY (north andeast) to OX’ and OY’.

with RT(θ) the transpose of R(θ). RT(θ) will sometimes be written R(−θ).

2.1 Rotation of the horizontal axes

Upon rotation of the horizontal measuring axes clockwise by angle θ’ as shown in Fig. 1,matrix [Zxx, Zxy; Zyx, Zyy] changes to [Z′xx, Z′xy; Z′yx, Z′yy] according to

[Z′xx Z′xyZ′yx Z′yy

]= R(θ′)

[Zxx ZxyZyx Zyy

]R(−θ′) (5)

Thus the elements of the second matrix are related to the first by the following equations:

Z′xx = (Zxx + Zyy)/2 + C sin(2θ′ + β) (6)

Z′xy = (Zxy − Zyx)/2 + C cos(2θ′ + β) (7)

Z′yx = −(Zxy − Zyx)/2 + C cos(2θ′ + β) (8)

Z′yy = (Zxx + Zyy)/2− C sin(2θ′ + β) (9)

whereC = [(Zxx − Zyy)

2 + (Zxy + Zyx)2]

12 /2 (10)

and β is defined bytan β = (Zxx − Zyy)/(Zxy + Zyx) (11)

83Magnetotelluric Tensor Decomposition: Insights from Linear Algebra and Mohr Diagrams


It is also useful to define an auxiliary angle β′ by

tan β′ = (Z′xx − Z′yy)/(Z′xy + Z′yx) (12)

withθ′ = (β′ − β)/2 (13)

and angle μ astan μ = (Zyy + Zxx)/(Zxy − Zyx) (14)

also ZL asZL = [(Zxx + Zyy)

2 + (Zxy − Zyx)2]

12 /2 (15)

Then plotting Z′xx against Z′xy as the axes are rotated (i.e. θ′ varies) defines a circle, known(with its axes) as a Mohr diagram. On such a figure axes for Z′yx and Z′yy may be included, todisplay the variation of these components also.

3. The depiction of MT tensors using Mohr diagrams

The Mohr diagram representation, straightforward for a 2 x 2 matrix, is an informative figurefor the student of linear algebra. For the MT case it may be a useful way to display results, aswill now be shown for the general case of 3D conductivity structure, and the particular casesof 2D and 1D conductivity structure to which the 3D case simplifies.

In this chapter in-phase and quadrature data will be presented in adjacent figures. There is anappeal in putting both in-phase and quadrature data on the same axes, as demonstrated bySzarka & Menvielle (1997) and Weaver et al. (2000), but when showing a full frequency rangeseparate sets of axes are practical. Also at times it is helpful to add axes for Z′yx and Z′yy as onFig. 2a, and such an addition is not possible for Z′xx and Z′xy axes common to both in-phaseand quadrature parts.

3.1 The general case; 3D structure

Mohr diagrams for the general case of the MT tensor are shown in Fig. 2a. Different pointson the diagram can be checked to confirm that Equations 6 to 15 for the rotation of axesare obeyed. The diagram is “Type 1” of Lilley (1998, p.1889), and shows how axes forZ′yx and Z′yy can be included. The diagram in Fig. 2a is drawn for relatively mild 3Dcharacteristics. Diagrams for strong 3D characteristics are discussed below and shown in anexample; however, what limits there may be to extreme 3D behaviour have not yet been fullyexplored. Some examples of 3D behaviour which appear to be prohibited will be discussed inthis chapter.

3.2 2D structure

When the geologic structure is 2D and the axes are rotated to be along and across geologicstrike, the MT tensor can be rotated to have the form [0, Zσ;−Zχ, 0], where it is expected bothZσ and Zχ are positive (a point discussed in Section 13.1 below). As shown in Fig. 2b, thediagram becomes a pair of circles with origins on the Z′xy axes, and angles μp and μq are zero.The in-phase and quadrature radial arms are parallel. Zσ and Zχ are known as the TE andTM modes (or vice-versa).



Z L

C

(Zxx+Zyy)/2

(Zxy-Zyx)/2

Z’xxp

Z’xyp

Z’xxq

Z’xyq

Z’xxp

Z’xyp

Z’xxq

Z’xyq

Zp χ Zp

σ Zq χ Zq

σ

Zp Zq

0 0

0 0

a

b

c

In-phase Quadrature

marks observed point (Zxy, Zxx)marks general point (Z’xy, Z’xx) after axes rotation of θ’

Z’xy

Z’xx

Z’yy

Z’yx

0

0

μ

β

β β

β’2θ’

Fig. 2. Mohr diagrams for a: The general MT tensor; both in-phase and quadrature parts takethis general form. b: The 2D case; radial arms are now parallel, with circle centres on thehorizontal axes. The 2D values Zσ and Zχ could be interchanged. c: The 1D case, for Zobserved as [0, Z;−Z, 0].

3.3 1D structure

If a 2D case simplifies further to become a 1D case, Zσ = Zχ = Z say, and the tensor for allrotations has the form [0, Z;−Z, 0]. The length C of the radial arm vanishes, and the diagramreduces to a pair of points on the horizontal axes, as shown in Fig. 2c.

4. Invariants of rotation of the measuring axes

It can be seen from Fig. 2a that the MT observations can be expressed in terms of seveninvariants of rotation. From different sets which are possible, and evident from formal analysis(Szarka & Menvielle, 1997; Weaver et al., 2000), this chapter adopts the invariants shownin Fig. 3. Thus the eight values of the MT tensor, all of which generally change upon axesrotation, become seven invariants plus one angle. That angle, θ′ in Fig. 2a, is the angle whichdefines the direction of the measuring axes (for example, with respect to north).



Z’xxp

Z’xyp

Z’xxq

Z’xyq

ZpL

ZqL

0 0

In-phase Quadrature

marks the observed points

μp μq

δβ

λp λq

Fig. 3. Seven invariants of rotation for a general 3D MT tensor.

4.1 Two invariants summarising the 1D character of the tensor

The two invariants ZLp,q summarising the 1D character are straightforward and were termed

“central impedances” by Lilley (1993). Note that the second could be expressed normalisedby the first, and so as a ratio (and then as an angle, taking the arctangent of that ratio).

4.2 Two invariants summarising the 2D character of the tensor

Two invariants λp,q measure the 2D character, and are also straightforward. They are naturallyangles, and were termed anisotropy angles by Lilley (1993). They may be expressed

λp,q = arcsin(Cp,q/ZLp,q) (16)

where λp,q is in the range 0 to 90◦. (Note this definition fails if Cp,q > ZLp,q and the relevant

circle encloses the origin.)

4.3 Two (of three) invariants summarising the 3D character of the tensor

Two angles μp,q characterising the 3D nature of the impedance tensor are also straightforward,and are shown in Fig. 3. It may be effective to express them as their mean and differencevalues, because certain mechanisms for causing 3D effects, especially static distortion, givethe same μ contribution to both the in-phase and quadrature parts of a tensor (Lilley, 1993). Insuch cases, static distortion of a regional 2D structure is then measured by (μq + μp)/2, andthe difference (μq − μp) would be zero. Thus (μq − μp), when non-zero, may be a measure ofany 3D effects present, beyond static distortion.

4.4 The third 3D invariant

A third 3D invariant (δβ = βq − βp) can be seen from Fig. 3 to be the angle by which the tworadial arms of the in-phase and quadrature circles are not parallel. It is significant as it alone, ofthe invariants chosen, links the in-phase and quadrature parts of an observed tensor. Anotherpossibility for this seventh invariant is (δβ − μq + μp), where the difference (μq − μp) isremoved first from δβ, and the departure of the radial arms from being parallel is then judgedafresh.



However a related angle for the seventh invariant, derived by Weaver et al. (2000), has thegreat utility that it appears again in the Mohr diagram for the phase tensor, to be introducedin Section 8 below. There, as angle γ, it has a simple significance concerning geologic strike.

4.5 Angles or their sines?

Weaver et al. (2000) take sines of the angles to give measures in the range 0 to 1, and thistechnique is adopted by Marti et al. (2005). However, if ambiguity is to be avoided, sucha procedure restricts the angles to being not greater than 90◦. Because examples occur forwhich the values of μp,q are greater than 90◦, it may be preferable to quote the invariants asangles, allowing them a 360◦ range (perhaps most usefully expressed in the range ±180◦).

5. Some basic techniques of linear algebra applied to the analysis of the MT tensor

To gain familiarity with the MT tensor, it is instructive to explore some common steps takenin matrix analysis. The steps described form part of the history of MT.

5.1 Separation into symmetric and antisymmetric components

The MT tensor Z, if split into symmetric Zs and antisymmetric Za parts

Z = Zs + Za (17)

may be written[Zxx ZxyZyx Zyy

]=

[Zxx (Zxy + Zyx)/2

(Zxy + Zyx)/2 Zyy

]+

[0 (Zxy − Zyx)/2

(Zyx − Zxy)/2 0

](18)

The second and antisymmetric part, Za, is immediately recognised as being of the ideal 1DMT form described in Section 3.3.

In much MT interpretation, it has been common practice to obtain “1D estimates” from 2Dor 3D data by taking the average of observed Zxy and (−)Zyx as (Zxy − Zyx)/2. Equation 18demonstrates the approximations made in such a procedure. For in taking such an average,it can be seen that the information in the first matrix term Zs is ignored, perhaps withoutjustification. Diagrammatically, the procedure is equivalent, in Fig. 4a, to representing thecircle on the left-hand side by the sum of the two circles on the right-hand side. The first circleon the right-hand side is then ignored, leaving the second circle which, reduced to its centralpoint, is a 1D case.

5.2 Separation into symmetric and 2D components

The exercise in Section 5.1 may be regarded as a separation into symmetric and 1Dcomponents, and suggests a similar separation of the observed tensor into two parts of whichthe second, Z2D, is chosen to be of ideal 2D form. The first part, Zs2, is found to again besymmetric.

Thus the MT tensor is expressedZ = Zs2 + Z2D (19)



a

b

=

=

+

+

Fig. 4. Separation of a 2 x 2 matrix, a: into symmetric and 1D parts as in Equation 18; b: intosymmetric and 2D parts as in Equation 20. The diagrams are drawn forZ = [1.75, 3.30;−0.70, 0.25]. Thus case a is: Z = [1.75, 1.3; 1.3, 0.25] + [0, 2.0;−2.0, 0], and caseb is: Z = [1.0, 0; 0, 1.0] + [0.75, 3.3;−0.7,−0.75].

and by partition as[Zxx ZxyZyx Zyy

]=

[(Zxx + Zyy)/2 0

0 (Zxx + Zyy)/2

]+

[(Zxx − Zyy)/2 Zxy

Zyx (−Zxx + Zyy)/2

](20)

where the second part is now of ideal 2D form.

Equation 20 is expressed in Mohr diagrams in Fig. 4b, where it can be seen that the first partis a point on the vertical axis, and the second part, Z2D, taken by itself plots as an ideal 2Dcircle with centre on the horizontal Z′xy axis. The intersections of the circle with the axis givethe values of the TE and TM impedances for this (now artificially) ideal tensor.

Thus the common practice with MT data, when seeking TE and TM values, of finding themaximum and minimum values of Z′xy as the observing axes are rotated, can be seen to betantamount to an assumption that the Zs2 part of the matrix be ignored, so that the circle forthe Z2D part indeed plots in ideal 2D form.

5.3 Eigenvalue analysis

Eigenvalues ζ1 and ζ2 of a matrix Z are found by solving the characteristic equation

ζ2 − (Zxx + Zyy)ζ + ZxxZyy − ZxyZyx = 0 (21)

to obtain

ζ1, ζ2 = (Zxx + Zyy)/2± [(Zxx + Zyy)2 + 4(ZxyZyx − ZxxZyy)]

12 /2 (22)

Three cases are possible and of interest. Each case will be discussed separately.



5.3.1 Eigenvalues are conjugate pairs

The first case occurs when

(Zxx + Zyy)2 + 4(ZxyZyx − ZxxZyy) < 0 (23)

and the two roots of the conjugate equation form a conjugate pair. The product of the tworoots, ζ1ζ2, will always be positive. An equivalent way of expressing Inequality 23 is as

(Zxx − Zyy)2 + 4ZxyZyx < 0 (24)

5.3.2 Eigenvalues are real and equal

The second case occurs when

(Zxx + Zyy)2 + 4(ZxyZyx − ZxxZyy) = 0 (25)

The two roots of the conjugate equation are now both real (positive or negative), and equal.In fact

ζ1 = ζ2 = (Zxx + Zyy)/2 (26)

Again, the product of the two roots, ζ1ζ2, will be positive.

5.3.3 Eigenvalues are real and different

The third case occurs when

(Zxx + Zyy)2 + 4(ZxyZyx − ZxxZyy) > 0 (27)

The two roots of the conjugate equation are now both real, different, and positive or negativedepending on the signs of (Zxx + Zyy) and (ZxxZyy − ZxyZyx), the trace and determinantrespectively of Z.

The product of the two eigenvalues is given by

ζ1ζ2 = det Z (28)

and is positive if detZ is positive, and negative if detZ is negative.

5.4 Eigenvalues on Mohr diagrams

The three eigenvalue cases discussed in the preceding three subsections are clearly definedwhen the data are plotted on Mohr diagrams. Eigenvalues which are real are showngraphically. The directions of their eigenvectors may be read from the diagrams remembering,in Fig. 2a, the 2θ′ anticlockwise rotation of the radial arm for, in Fig. 1, the θ′ clockwise rotationof the axes.

Thus for the case of Section 5.3.1, circles which (as in Fig. 5a) do not touch the vertical axesobey Inequality 23. Their eigenvalues are complex conjugate pairs, and real eigenvectors forthem do not exist.

Secondly, for the case discussed in Section 5.3.2, Fig. 5b shows a circle which is just touchingboth vertical axes, Z′xy = 0 and Z′yx = 0. The eigenvalues may be read off the Z′xx axis, as



a b c

d e f

Fig. 5. Eigenvector directions (according to the radial arrows) and eigenvalues (where thearrowheads touch an axis) displayed on Mohr diagrams for a 2 x 2 matrix. a: Eigenvalues arecomplex conjugates, and are not evident on the diagram. b: Eigenvalues are real and equal;both eigenvectors are the same. c: Eigenvalues are real and different; eigenvectors are notorthogonal. d: Eigenvalues are real and different; eigenvectors are orthogonal. e: AssociatedMT eigenvalues are real and different; eigenvectors are not orthogonal. f: Associated MTeigenvalues are real and different; eigenvectors are orthogonal (the 2D case). In Fig. 5a axesare also drawn for Z′yx and Z′yy. Between the Z′xx and Z′yy axes, the product Z′xy.Z′yx < 0. Onthe Z′xx and Z′yy axes, Z′xy.Z′yx = 0. Outside the Z′xx and Z′yy axes, Z′xy.Z′yx > 0.

(Zxx + Zyy)/2. The direction of the repeated eigenvector corresponds to the direction of aradial arm which is horizontal in the diagram, as shown.

Thirdly, for the case discussed in Section 5.3.3, in Fig. 5c a circle is shown which, like examplesto be discussed below, now crosses the vertical axes. The two eigenvalues may again be readoff the Z′xx axis where the circle cuts this axis, and the two eigenvector directions are given bythe θ′ values for the radial arms to these points.

The example in Fig. 5c demonstrates that real eigenvalues correspond to the MT case of “phasegoing out of quadrant”. Also it can be seen, by an extension of the discussion in Section 5.3.3above, that for a Mohr circle to not capture the origin the product of the two eigenvalues mustbe positive; i.e. detZ must be positive.

Of equal interest in MT are the eigenvectors of the associated problem, which correspond tointersections of a circle with the horizontal Z′xy axis. As shown in Fig. 5e, these might beregarded as an approximation to 2D TE and TM values, and for the true 2D case (Fig. 5f) theywill indeed be so. These eigenvalues may be found by similar formalism, or by elementarytrigonometry based on Fig. 2a. They may be evident from inspection, as in Lilley (1993).



6. Singular value decomposition

It is instructive also to apply SVD to an MT tensor, taking the in-phase and quadrature partsof the tensor separately.

As described for example by Strang (2005), decomposition by SVD factors a matrix A into

A = UΣVT (29)

where the columns of V are eigenvectors of ATA, and AT denotes the transpose of A. Thecolumns of U (which are eigenvectors of AAT) may be found by multiplying A by thecolumns of V. The singular values on the diagonal of Σ are the square roots (taken positiveby convention, a most important point in the present context) of the non-zero eigenvalues ofAAT. As Σ is diagonal, it is straightforward to convert it to the antidiagonal form of an ideal2D tensor, for example by expressing Equation 29 as

A = UΣWW−1VT (30)

where W = [0, 1;−1, 0] and VT is pre-multiplied by W−1, with W−1 denoting the inverse ofW.

Equations for this form of the SVD of an MT tensor were derived in an earlier paper (Lilley,1998). Taking phase relative to the H signal, so that Hq is zero, Equation 1 is written in itsin-phase and quadrature parts as

Ep,q = R(−θep,q)

[0 Υp,q

−Ψp,q 0

]R(θhp,q)H (31)

where the electric and magnetic observation axes are rotated clockwise independently, theelectric axes by θep,q and the magnetic axes by θhp,q. Thus the in-phase part of the tensor, Zp,is factored into [

Zxx p Zxy pZyx p Zyy p

]= R(−θep)

[0 Υp−Ψp 0

]R(θhp) (32)

where

θep =12

[arctan

Zyy p − Zxx p

Zxy p + Zyx p+ arctan

Zyy p + Zxx p

Zxy p − Zyx p

](33)

θhp =12

[arctan

Zyy p − Zxx p

Zxy p + Zyx p− arctan

Zyy p + Zxx p

Zxy p − Zyx p

](34)

Υp −Ψp = cos(θep + θhp)[(Zxy p + Zyx p)− tan(θep + θhp)(Zxx p − Zyy p)] (35)

andΥp + Ψp = cos(θep − θhp)[(Zxy p − Zyx p) + tan(θep − θhp)(Zxx p + Zyy p)] (36)

(Equation 36 corrects equation 22 of Lilley (1998), where a negative sign is missing.) A similarset to Equations 32, 33, 34, 35 and 36 applies for the quadrature part of a tensor, with subscriptq replacing p.

The quantities Υp,q and Ψp,q are termed principal values. An examination of the possibilitythat one or both of Ψp,q may be zero or negative is addressed in Section 13.3.



observed point (Zxxp =-1, Zxyp =7)

Z’xyp

Z’xxp

0 5.5

1

θep - θhp

2θep

Υp

Ψp

u1 v1

u2 v2

Ex Ex’

Ey

Ey’

Hx Hx’

Hy

Hy’

θep

θ hp

Fig. 6. Diagram showing the Mohr representation and the SVD of the matrix in Equation 37.The values of θep, θhp, Υp and Ψp are evident as 32◦, 21◦, 8.1 and 3.1. The axes for E′x, E′y andH′x, H′y show the rotations, from Ex, Ey and Hx, Hy by θep and θhp respectively, to give anideal 2D antidiagonal response.

The quantities arising from the SVD may be displayed on a Mohr diagram as in Fig. 6, whichhas been drawn for a tensor

Zp = [−1, 7;−4, 3] (37)

The values of θep, θhp, Υp and Ψp may be evaluated by the equations above as 31.7◦, 21.4◦, 8.09and 3.09 respectively. These values may also be read off the figure.

When the matrix of Equation 37 is put into a standard computing routine, the SVD returnedis commonly in the form of Equation 29:

Zp =

[−.8507 −.5257−.5257 .8507

] [8.0902 0

0 3.0902

] [.3651 −.9310−.9310 −.3651

](38)

which may be given the form of Equation 32 by expressing it first as

Zp =

[.8507 −.5257.5257 .8507

] [0 8.0902

−3.0902 0

] [.9310 .3651−.3651 .9310

](39)



and then as

Zp =

[cos 31.71◦ − sin 31.71◦sin 31.71◦ cos 31.71◦

] [0 8.0902

−3.0902 0

] [cos 21.41◦ sin 21.41◦− sin 21.41◦ cos 21.41◦

](40)

The columns of the first matrix [cos 31.71◦,− sin 31.71◦; sin 31.71◦, cos 31.71◦], whichhave been derived from U in Equation 29, can be seen to define unit vectors u1and u2 which are as given for E′x and E′y in Fig. 6. The rows of the thirdmatrix [cos 21.41◦, sin 21.41◦;− sin 21.41◦, cos 21.41◦], which have been derived from VT inEquation 29, can be seen to define unit vectors v1 and v2 as given for H′x and H′y in Fig. 6.Also, as is evident, the singular values in the diagonal of matrix Σ (the second matrix) givethe values of Υp and Ψp.

Note that by rotation of the electric and magnetic axes separately, the MT tensor has beenreduced to an ideal 2D form. In a search for the nearest 2D model in a 3D situation the resultsof SVD may be valuable to bear in mind; the magnetic axes may be an indication of regionalstrike, with the electric axes showing the brunt of the distortion. A disadvantage of such ananalysis however is that a rather artificial conductivity structure is required to simply twistthe electric field at an observing site to explain the different rotations required of the E andH axes. A more specific (if simple) model for local distortion is widely accepted, and will bediscussed in Section 11. With this model, in the case of near-singular MT data, it will be shownthat a surficial strike direction is determined. However regional strike determination remainspossible if regional anisotropy is high.

7. A condition number to measure singularity in an MT tensor

It is common experience to find very strong anisotropy in an observed tensor, both fordistorted 2D cases, and indeed generally. As a consequence, the tensor approaches a conditionof singularity, in both its in-phase and quadrature parts. In a Mohr diagram the condition ofsingularity is shown by a circle touching the origin. If for example Zp is a singular tensor, thenthere is some rotation of axes for which both Z′xx p and Z′xy p

are zero (or indistinguishable from

zero, when error is taken into account). For the student of linear algebra, an example of a nullspace occurs: it is the line of the direction of nil electric field change, holding for all magneticfield changes.

A condition number may be used to warn that singularity is being approached (Strang,2005). When the condition number becomes high in some sense, the matrix is said to beill-conditioned (Press et al., 1989). The condition number suggested by Strang (2005) is thenorm of the matrix (sometimes called the spectral norm) multiplied by the norm of the inverseof the matrix; or equivalently, the greater principal value of the matrix divided by the lesserprincipal value. For the 2 x 2 matrix Z in Equation 1 the condition numbers κp,q are

κp,q = Υp,q/|Ψp,q| (41)

The greater and lesser principal values Υp,q and Ψp,q are given by Equations 35 and 36 and, asdiscussed above, are the singular values of the matrix. Following the convention that singularvalues are never negative, a modulus sign is put into the denominator of Equation 41 to covercases where computation of the lesser principal value produces a negative number. In terms



0

10

20

30

40

50

60

70

80

90

100

110

120

130

κ p,q

0 10 20 30 40 50 60 70 80 90

λp,q

0.1

1

10

100

1000

10000

100000

κ p,q

0 10 20 30 40 50 60 70 80 90

λp,q

Fig. 7. The condition number κp,q displayed as a function of the anisotropy angle λp,q indegrees, using linear (left) and logarithmic (right) scales for κp,q.

of the Mohr representations in Figs 2, 3 and 6, from Equation 41 the condition numbers aregiven by

κp,q = (ZLp,q + Cp,q)/|(ZL

p,q − Cp,q)| (42)

that isκp,q = 2/(cscλp,q − 1) + 1 (43)

remembering Equation 16. Fig. 7 shows κp,q as a function of λp,q. While λp,q itself is ameasure of condition, Fig. 7 demonstrates that κp,q is more sensitive than λp,q as ill-conditionis approached. Note that κp,q ≥ 1.

An example is given in Fig. 8 of condition numbers determined for a set of data from a recentMT site, NQ142, in north Queensland, Australia. An increase in condition number above10 monitors the decrease of the lesser principle value Ψp,q, which becomes negligible. Thevariation of angle θep,q with period (T) stabilizes as condition numbers rise, an effect discussedbelow in Section 11.

8. The phase tensor

Caldwell et al. (2004) introduced a “phase tensor” based on the magnetotelluric tensor, andthe concept was further developed by Bibby et al. (2005). The phase tensor is a real matrix Φ,defined by

Φ = Z−1p Zq (44)

and it has the property that it is unaffected by the in-phase distortion (described in Section 11.1below) which is recognised as common in MT data. Note, however, that the computation ofphase tensor values may encounter difficulties for high condition numbers and singularitiesin Zp and Zq, which can be caused by strong distortion.

8.1 Mohr diagram for the phase tensor

The phase tensor, a 2 x 2 matrix, can also be represented by a Mohr diagram, as described byWeaver et al. (2003; 2006). The general form is shown in Fig. 9, following the convention foraxes of Fig. 2a. Equations 6 to 15 can be adapted to apply to the quantities shown in Fig. 9.



100

101

102

103

104

105

κ p

10-4 10-3 10-2 10-1 100 101 102 103 104 105

T (s)

100

101

102

103

104

105

κ q

10-4 10-3 10-2 10-1 100 101 102 103 104 105

T (s)

10-2

10-1

100

101

102

103

Ψp

, Υp

10-2

10-1

100

101

102

103

Ψq

, Υq

-90

0

90

θhp

-90

0

90

θhq

-90

0

90

θep

-90

0

90

θeq

-50

0

50

Z’x

x p T

1/2

-50 0 50 100Z’xyp T

1/2

-50

0

50

Z’x

x q T

1/2

-50 0 50 100Z’xyq T

1/2

Fig. 8. The NQ142 data plotted as Mohr diagrams and analysed by SVD to give: angles θep,θeq, θhp and θhq; the principal values Υp, Ψp, Υq and Ψq; and the condition numbers κp andκq. The variation of period with colour in the circles is the same as for the other plots. TheMohr diagrams follow the form of Fig. 2a, and have their impedance values multiplied bythe square root of period (T) to make the plots more compact. Where circles enclose theorigin, their lesser principal value (Ψ) is given an artificially high value to flag thiscircumstance; however condition numbers are computed nevertheless.



Φ’xx

Φ’xy

(Φxx+ Φyy)/2

(Φxy- Φyx)/2

0

γ2θ’

marks observed point

marks general point after axes rotation

strike direction based on maximum Φ’xx

strike direction based on Φmax

J1

J2

J3

Fig. 9. Diagram showing the Mohr representation of the general phase tensor. Axes for Φ′yyand Φ′yx could be added (following Fig. 2a). The principal values (equivalent to Υp and Ψp inFig. 6) are given by (J1

2 + J32)1/2 ± J2, and are denoted Φmax and Φmin by Caldwell et al.

(2004).

The angle γ, which is a measure of 3D effects, is the seventh invariant of Weaver et al. (2000),described in Section 4.

The regional 2D strike direction chosen on phase considerations, as advocated by Bahr (1988)for example, is given by the point of maximum Φ′xx, which is the highest point of the circleas drawn. However when the regional structure is recognised as being 3D, the best estimateof geologic strike is recommended by Caldwell et al. (2004) as the orientation of the principalaxes of the phase tensor, shown in Fig. 9 as “strike direction based on Φmax”.

Note that a Mohr diagram for the phase tensor shows the variation of phase-tensor values asthe observing axes are rotated, but that these values are generally not those of the usual MTZxy phase.

Also note that a Zxy phase going “out of quadrant” does not imply that a principal value of Φ

is negative. In fact detΦ is expected to be never negative, consistent with the principal valuesof Φ (Φmax and Φmin) being never negative.

Examples of Mohr diagrams for phase tensors, which illustrate some of these points, are givenin Lilley & Weaver (2010).



8.2 Mohr diagram for the 2D phase tensor

For the 2D case, the Mohr diagram reduces to a circle with centre on the Φ′xx axis. The TE andTM phase values are then the arctangents of the intercepts which, from the analysis alreadygiven, are eigenvalues of the phase tensor. The two eigenvectors are orthogonal.

8.3 Mohr diagram for the 1D phase tensor

For the 1D case, the Mohr diagram reduces to a point on the Φ′xx axis, marking the tangent ofthe phase value of the 1D MT response.

9. Invariants for the general phase tensor

To characterize the phase tensor just three invariants of rotation are needed, as pointed out byCaldwell et al. (2004), together with an angle defining the direction of the original observingaxes. Options for invariants are evident from an inspection of Fig. 9, and include thoseadopted in Section 4 for Fig. 3. Weaver et al. (2003) chose J1, J2 and J3 as shown in Fig. 9,which neatly summarize 1D, 2D and 3D characteristics, respectively.

10. Some complications illustrated by Mohr diagrams

10.1 Conditions for Zxy and Zyx phases out of quadrant

For simple cases of induction in 1D layered media, Zxy phases will be in the first quadrant, andZyx phases in the third quadrant. However, it is not uncommon with complicated geologicstructure to find phases which are out of these expected quadrants, for some orientation ofthe measuring axes. Experimental or computational error may be invoked in explanation,when in fact a common cause is simply distortion. Such distortion may be understood byreference to Fig. 3. Clearly once, due say to distortion, angle μp (or μq) increases to the pointwhere the in-phase (or quadrature) circle first touches and then crosses the vertical Z′xx axis,for an appropriate direction of the observing axes the phase observed of Zxy will be “out ofquadrant”.

Adopting, in the rotated frame, the common definition for phase φ′xy of

φ′xy = arctan(Z′xyq/Z′xy p

) (45)

where the signs of numerator and denominator are taken into account, then φ′xy is in the firstquadrant when both Z′xy p

and Z′xyqare positive.

The condition for the phase of Z′xy to go out of quadrant is that one or both of its in-phase andquadrature parts should be negative. On the Mohr circle diagram, this condition is equivalentto either or both of the in-phase and quadrature circles for the data crossing the vertical axis.Phases greater than 90◦ are then possible for Z′xy. If the circles however remain to the rightof their vertical axes, the Z′xy phase will be in quadrant for any orientation of the measuringaxes.

With reference to Fig. 3, for the crossing of the vertical axis to occur, it can be seen that λp,qand μp,q must together be greater than 90◦, i.e.

λp,q + |μp,q| > 90◦ (46)



With reference to the Z′yy pand Z′yx p

axes also included in Fig. 2a, consequences for the phases

of other elements are also clear. The symmetry of the figure is such that if the circle crossesthe Z′xx p axis to the left, it will also cross the Z′yy p

axis to the right. Thus if phases out of

quadrant are possible for Z′xy, so are they also possible for Z′yx. However, the latter will occurfor rotations 90◦ different from the former.

Similar considerations will give rules regarding the quadrants to be expected for Z′xx and Z′yyphases. An analysis of Fig. 2a shows that these phases will generally change quadrant witha rotation of the observing axes. An exception, when they would not do so, would be if thecircle in Fig. 2a did not cross the horizontal Z′xy p

axis, but stayed completely above (or below)

this axis (and the circle for the quadrature part of the tensor behaved similarly).

10.2 Strike coordinates from phase considerations

The situation of Zxy phase out of quadrant (for example for the part of the circle to theleft of the vertical axis in Fig. 5c) causes difficulty in the symmetric-antisymmetric and thesymmetric-2D partitions of Sections 5.1 and 5.2. First, if the circle in Fig. 5c is simply moveddown so that its centre lies on the horizontal axis, the left-hand intercept with the horizontalaxis is less than zero. The circle has enclosed the origin, and a negative value is obtained forthe lesser of TE and TM, contrary to expectation (see Section 13.1).

Similarly there are methods which seek, as geologic strike coordinates, those given bymaximum and minimum Z′xy phase values when the axes are rotated. These methods willfind erratic phase behaviour when either or both of the in-phase and quadrature circles crossthe vertical axis as in Fig. 5c.

However Z′xy phase will usually be at or near a maximum (or minimum) at the right-handside of a circle which on its left-hand side crosses the vertical axis. Thus sensible results formaximum or minimum phase can be expected, if based only on a determination from theright-hand side.

11. Geologic interpretation of angles found by SVD, when condition numbers arehigh

Singular value decomposition of observed magnetotelluric tensors, taking the in-phase andquadrature parts separately, often produces directions which are frequency independentbelow a certain frequency which is found to be common to both in-phase and quadratureparts. This section examines how such directions may be related to the angles which arise inbasic distortion models of MT data. It is found that while MT data are expected in general tobe frequency dependent, there are particular limiting cases which are constant with frequency.Understanding such observed data may be useful in interpretation.

11.1 The Smith (1995) model for local distortion

This section applies the Smith (1995) description of local “static” distortion, which followsBahr (1988) and Groom & Bailey (1989). In the absence of local distortion the measuredmagnetic field H is related to the regional electric field Er by the tensor Zr:

Er = ZrH (47)



and with distortion d present, affecting the measured electric field Em only,

Em = dEr (48)

andEm = ZmH = dZrH (49)

For the case of 2D regional structure, with measuring axes aligned with regional strike,

Zr = [0, Zr12; Zr

21, 0] (50)

and soEm = d[0, Zr

12; Zr21, 0]H (51)

where the overscore˜indicates observing axes aligned with regional strike.

If the observation axes are north-south and east-west, say, and need to be rotated angle θ foralignment with regional strike, then Equation 51 (without distortion) gives

R(θ)

[Em

xEm

y

]=

[0 Zr

12Zr

21 0

]R(θ)

[HxHy

](52)

and with distortion

R(θ)

[Em

xEm

y

]= d

[0 Zr

12Zr

21 0

]R(θ)

[HxHy

](53)

In co-ordinates aligned with the surficial geology the distortion amounts to scaling the electricfields by different amounts, g1 and g2, in directions parallel and perpendicular to the surficialstrike (Zhang et al., 1987). Distortion matrix d then has the form[

d11 d12d12 d22

]= R(−αs)

[g1 00 g2

]R(αs) (54)

where due to the restricted distortion model the d21 element has the value d12, g1 and g2are real constants (frequency independent), and αs is the angle from the regional strikecoordinates to the strike coordinates of the surficial structure causing the distortion. Thedecomposition of d in Equation 54 is displayed in Fig. 10.

Substituting the expression for d from Equation 54 into Equation 53 gives

Em = R(−θ − αs)

[g1Zr

21 sin αs g1Zr12 cos αs

g2Zr21 cos αs −g2Zr

12 sin αs

]R(θ)H (55)

At this stage the in-phase and quadrature parts can be considered separately. Avoiding theencumbrance of additional notation, now assume that just the in-phase part of the measuredE field is being considered, giving information on just the in-phase parts of the regionalimpedance tensor Zr.

Then following the form of Equation 32, SVD can be carried out on the central tensor inEquation 55, expanding it as[

g1Zr21 sin αs g1Zr

12 cos αsg2Zr

21 cos αs −g2Zr12 sin αs

]= R(−ηe)

[0 Υ−Ψ 0

]R(ηh) (56)



d’11

d’12g1 g1

g2 g2

cos αs cos αs2αs

- sin αs sin αs0 0 0 0

2 2

2 2 =

Fig. 10. Mohr diagrams for the decomposition in Equation 54. On the left the diagram showsthe distortion matrix d. It is positive definite, due to the assumption of strictly 2D surficialanisotropy, and so has orthogonal eigenvectors. The radius of the circle is (g2 − g1)/2. Thenumerical matrix taken for the example is [3.5, 2.6; 2.6, 6.5], and the directions andmagnitudes of its eigenvectors are shown by rotating the radial arm until it is parallel to thevertical axis. The eigenvalues then are g1 = 2 and g2 = 8. The angle 2αs on the figure is 60◦,indicating that a rotation of axes anticlockwise through 30◦ is necessary to change the matrixto [2, 0: 0, 8]. On the right is shown the matrix decomposition as in Equation 54. The threediagrams represent the three matrices on the right-hand side of that equation, in turn. Thefirst Mohr diagram represents a rotation by angle (−αs). The second Mohr diagram issymmetric, representing a positive definite matrix according to the scaling factors g1 and g2.The third matrix is like the first, representing a rotation by angle (+αs). Note the changes ofscale between the diagrams.

where the angles ηe and ηh, and the principal values Υ and Ψ are given by

ηe + ηh = arctan[−g2Zr

12 sin αs − g1Zr21 sin αs

g1Zr12 cos αs + g2Zr

21 cos αs

](57)

ηe − ηh = arctan[−g2Zr

12 sin αs + g1Zr21 sin αs

g1Zr12 cos αs − g2Zr

21 cos αs

](58)

Υ + Ψ = [(g1Zr21 sin αs − g2Zr

12 sin αs)2 + (g2Zr

21 cos αs − g1Zr12 cos αs)

2]1/2 (59)

andΥ−Ψ = [(g1Zr

21 sin αs + g2Zr12 sin αs)

2 + (g1Zr12 cos αs + g2Zr

21 cos αs)2]1/2 (60)

Replacing into Equation 55 the matrix in its expanded form as in Equation 56, and combiningadjoining rotation matrices, gives

Em = R(−θ − αs − ηe)

[0 Υ−Ψ 0

]R(θ + ηh)H (61)

Thus in the SVD of an MT tensor resulting from 2D surficial distortion of a regional 2Dstructure, the regional strike θ and the various distortion quantities g1, g2 and αs occurin a way which does not allow their straightforward individual solution, because the SVDproduces values for (θ + αs + ηe) and (θ + ηh). It is of interest however to examine two limitingcases, as in the following sections.



11.2 First limiting case

Now consider, in Equations 57, 58, 59 and 60, that g2 � g1, so that terms in g1 are ignoredwith respect to terms in g2. The results are then obtained that

Υ = g2[(Zr12 sin αs)

2 + (Zr21 cos αs)

2]1/2 (62)

andΨ = 0 (63)

For the angles ηe and ηh the limiting case gives

ηe = 0 (64)

andηh = − arctan(Zr

12tan αs/Zr21) (65)

While in Equation 65 for ηh the values Zr12, Zr

21 and αs still occur together, for ηe there is thesimple result of zero. Fed back into Equation 61 it is seen that SVD of an MT tensor now gives,in the equivalent of θe in Equation 31, a result for (θ + αs): that is, the direction of surficial 2Dstrike.

Further, note that if Zr12/Zr

21 = −1, then ηh = αs, and the magnetic axes are also alignedwith (or normal to) the surficial strike. This case can be seen to be that where the regional 2Danisotropy is out-weighed by the surficial anisotropy, so that the former approximates a 1Dcase.

11.3 Second limiting case

A second limiting case might be gross inequality in the TE and TM components of the regional2D impedance. Consider the case where |Zr

12| � |Zr21|, so that terms involving the latter may

be ignored with respect to terms involving the former. Then Equations 57, 58, 59 and 60 give

Υ = Zr12[(g2 sin αs)

2 + (g1 cos αs)2]1/2 (66)

andΨ = 0 (67)

For the angles ηe and ηh this limiting case gives

ηe = − arctan(g2tan αs/g1) (68)

andηh = 0 (69)

Fed back into Equation 61, it is seen that SVD of an MT tensor for the case where ηh = 0 nowgives, in the equivalent of θh in Equation 31, a result for θ: that is, the direction of regional 2Dstrike.

12. Discussion and example

The two limiting cases, where realised, give angles of interest. Also, it should be noted thatboth cases correspond to the MT tensor being singular, or nearly so.



Z’xxp

Z’xyp

Z’xxq

Z’xyq0 0

In-phase Quadrature

marks "observed" points

β β

Fig. 11. Hypothetical example of a 2D case with the in-phase and quadrature radial armsanti-parallel. Such cases appear to be prohibited in nature, but inadvertently can be posed asnumerical examples.

The example in Fig. 8, for frequencies less that 10 Hz, gives θep,q values which are fixed at−67◦, while θhp,q values continue to vary with period. The model of Section 11.1 is therefore agood candidate for the interpretation of the data. Applying the results of Section 11.2, surficialstrike is determined at −67◦ (±90◦ allowing for the usual ambiguity).

However the observation that there is a critical frequency, say fc, only below which θep,qvalues are frequency independent (and in the case of Fig. 8 are stable at −67◦), implies acontradiction to the initial distortion model in Section 11.1, which was frequency independent,generally. The existence of such a “critical frequency” suggests distortion which is “at adistance”, rather than immediately local.

13. Some remaining problems

13.1 TE and TM modes both positive?

It is well known that for the 2D case the TM mode is always positive (Weidelt & Kaikkonen,1994), but proof has not been achieved for the TE case. It can be seen from Fig. 2b that werethe TE case to be negative in either in-phase or quadrature part (or both) then the appropriatecircles would enclose their origin of axes. Thus a general proof that circles cannot enclosetheir origins (discussed in Section 13.3 below) would also prove that the TE mode can neverbe negative.

13.2 Radial arms anti-parallel

The 2D example given in Fig. 2b, where the in-phase and quadrature radial arms are parallel,represents the common case observed without exception in the experience of the presentauthor. However, in principle the case shown in Fig. 11 where the in-phase and quadratureradial arms are anti-parallel would also be two dimensional. If there is a theoretical reasonwhy cases with radial arms anti-parallel as in Fig. 11 do not occur in practice it would be anadvance to have it clarified.



observed point (Zxxp =-3, Zxyp =3)

Z’xyp

Z’xxp

0 4

1

Υ p

Ψ p

θep - θhp

2θep

u1v1u2

v2

Ex

Ex’

Ey

Ey’ Hx Hx’

Hy

Hy’

θep

θhp

Fig. 12. The Mohr diagram for the matrix in Equation 70. The circle encloses the origin ofaxes because the matrix has a negative determinant. The values of θep, θhp, Υp and Ψp areevident as 51◦, 25◦, 6.3 and −1.8 (Ψp being of negative sign when the direction of Υp from theorigin defines positive).

13.3 Circles capturing the origin: a note on negative determinants

It is also observed that circles do not capture or enclose their origins of axes, except for reasonsof obvious error, or usual errors associated with singularity. The author suggests that thisobserved behaviour corresponds to the prohibition of components of negative resistivity in theelectrical conductivity structure. Again a proof regarding this possibility would be welcome.

Negative determinants (no matter how they arise) need care in their analysis, as the followingexample illustrates. If, in contrast to the example in Section 6, the determinant (say of Zp) isnegative, then the matrices ZpZT

p and ZTp Zp will each have a negative eigenvalue. However

in a conventional SVD these negative eigenvalues are taken as positive singular values, andto allow for this action the sign is also changed of the eigenvector of U which they multiply.

To demonstrate this point, consider the hypothetical matrix

Zp = [−3, 3;−1, 5] (70)

which has a negative determinant. The Mohr diagram representation of the matrix is shownin Fig. 12.



By Equations 33, 34, 35 and 36, the values of θep, θhp, Υp and Ψp are evaluated as51.26◦, 24.70◦, 6.36 and−1.88 respectively. As before, by multiplying out the right-hand side, itcan be checked that these values do indeed satisfy Equation 32. Also these values may be readoff Fig. 12 as 51◦, 25◦, 6.3 and −1.8, but now difficulties arising from the negative determinantare evident. Taking the direction of Υp from the origin to define positive, the value of Ψp isseen to be negative.

Put into a standard computing routine, the SVD returned for the matrix in Equation 70 is

Zp =

[.6257 .7800.7800 −.6257

] [6.3592 0

0 1.8870

] [−.4179 .9085−.9085 −.4179

](71)

which may also be obtained formally by determining the eigenvalues and eigenvectors ofZT

p Zp. To give it the form of Equation 32, Equation 71 may be expressed

Zp =

[.6257 .7800.7800 −.6257

] [0 6.3592

−1.8870 0

] [.9085 .4179−.4179 .9085

](72)

and then written

Zp =

[cos 51.26◦ sin 51.26◦sin 51.26◦ − cos 51.26◦

] [0 6.3592

−1.8870 0

] [cos 24.70◦ sin 24.70◦− sin 24.70◦ cos 24.70◦

](73)

The rows of the third matrix [cos 24.70◦, sin 24.70◦;− sin 24.70◦, cos 24.70◦] define unit vectorsv1 and v2 for H′x, H′y as shown in Fig. 12. As for Fig. 6, these indicate a rotation of the Hx, Hyobserving axes.

However, the columns of the first matrix [cos 51.26◦, sin 51.26◦; sin 51.26◦,− cos 51.26◦] defineunit vectors u1 and u2 also as shown in Fig. 12. These unit vectors do not now represent justa simple rotation of the Ex, Ey axes, but also the reflection of one of them. The negative valueof Ψp given by Equations 35 and 36 has been cast by the SVD analysis as a positive singularvalue in the second matrix in Equation 71. To compensate for this convention, the directionof one of the axes (originating as the direction of an eigenvector of U in Equation 29) hasbeen reversed. Any model of distortion rotating the electric fields is thus contravened. Anextra physical phenomenon is required to explain the reflection of one of the rotated electricfield axes. Pending such an explanation, a negative principal value (−1.8870 in the presentexample) must be regarded with great caution.

In normal SVD formalism the change of sign of an eigenvector, as demonstrated in the aboveexample, may occur with little comment. In the present case however it has had the profoundeffect of destroying, after rotation, the “right-handedness” of the observing axes of the MTdata.

14. Conclusion

Basic 2 x 2 matrices arise commonly in linear algebra, and Mohr diagrams have wideapplication in displaying their properties. Complementing the usual algebraic approach, theymay help the student understand especially anomalous data.

Such Mohr diagrams have proved useful in checking MT data for basic errors arising inthe manipulation of time-series; in checking hypothetical examples for realistic form; andin checking for errors which are demonstrated by negative determinants.



Invariants of an observed tensor under axes rotation are evident from inspection of such Mohrdiagrams. The invariants may be used as indicators of the geologic dimensionality of thetensor; they may also be useful in the inversion and interpretation of MT data. Mohr diagramsof the MT phase tensor also show invariants which measure geologic dimensionality, andindicate regional geologic strike.

Attention has been given to monitoring the singularity of an MT tensor with a conditionnumber, as tensors which approach singularity are common. Carrying out SVD on observeddata with high condition numbers can indicate a direction of surficial 2D strike, for the simplecase of the surficial distortion of a regional 2D MT response. In some circumstances the 2Dregional strike is found in this straightforward way.

Particularly noteworthy is the observation of a critical frequency below which an observedMT tensor becomes sufficiently ill-conditioned for the surficial strike to become evident.The example in Fig. 8 shows this behaviour to apply at frequencies below 10 Hz. That afrequency-independent model produces a frequency-dependent result in this way requirescare in interpretation.

Remaining problems for which proofs would augment the subject are the evident prohibitionof negative TE values, the evident non-observation of antiparallel radial arms, and the evidentnon-observation of MT tensor data with negative determinant values (corresponding to Mohrcircles prohibited from enclosing their origins of axes).

15. Acknowledgements

The author thanks Peter Milligan, Chris Phillips and John Weaver for many valuablediscussions over recent years on the subject matter of this chapter. Data for the NQ142example were provided by Geoscience Australia.

16. References

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J. Int. 158: 457 – 469.Eggers, D. E. (1982). An eigenstate formulation of the magnetotelluric impedance tensor,

Geophysics 47: 1204 – 1214.Groom, R. W. & Bailey, R. C. (1989). Decomposition of magnetotelluric impedance tensors

in the presence of local three-dimensional galvanic distortion, J. Geophys. Res.94: 1913 – 1925.

Gubbins, D. & Herrero-Bervera, E. (eds) (2007). Encyclopedia of Geomagnetism andPaleomagnetism, Encyclopedia of Earth Sciences, Springer.

Hobbs, B. A. (1992). Terminology and symbols for use in studies of electromagnetic inductionin the Earth, Surv. Geophys. 13: 489 – 515.

LaTorraca, G. A., Madden, T. R. & Korringa, J. (1986). An analysis of the magnetotelluricimpedance for three-dimensional conductivity structures, Geophysics 51: 1819 – 1829.



Lilley, F. E. M. (1993). Magnetotelluric analysis using Mohr circles, Geophysics 58: 1498 – 1506.Lilley, F. E. M. (1998). Magnetotelluric tensor decomposition: Part I, Theory for a basic

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magnetotelluric tensor invariants and the phase tensor of Caldwell, Bibby andBrown, in J. Macnae & G. Liu (eds), Three-Dimensional Electromagnetics III, number 43in Paper, ASEG, pp. 1 – 8.

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